L(s) = 1 | + (−0.866 − 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (0.499 + 0.866i)21-s + (1.73 − i)23-s + (0.5 − 0.866i)25-s − 0.999i·27-s + (−0.5 − 0.866i)29-s + i·31-s + (0.5 − 0.866i)37-s + (0.866 − 0.499i)39-s + (0.5 − 0.866i)41-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (0.499 + 0.866i)21-s + (1.73 − i)23-s + (0.5 − 0.866i)25-s − 0.999i·27-s + (−0.5 − 0.866i)29-s + i·31-s + (0.5 − 0.866i)37-s + (0.866 − 0.499i)39-s + (0.5 − 0.866i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7937448877\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7937448877\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - iT - T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + iT - T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 - iT - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 - iT - T^{2} \) |
| 83 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.395794327973724572542950809297, −8.428173491072348844807842524811, −7.44574116559945077173051130179, −6.85352636244171981249964815409, −6.27078246370046160692075894621, −5.34061011690685841305512665707, −4.49175921636616920517490711343, −3.49257383848948868118081263081, −2.24406348362199057281227418974, −0.888242199047591667072394774817,
0.984191250969131093375267666571, 2.95700914505775444901886205395, 3.39310884677969315616905081443, 4.94403320981449114731037742044, 5.24721955480935101847446320199, 6.11958908952040342168920661993, 7.05400669287435484507416342560, 7.59063404715677768496942707807, 9.003120812012892120012643006380, 9.516412455389990836658896422534